The Wellbuilt Company produces two types of wood chippers, Deluxe and Economy.
The Deluxe model requires 3 hours to assemble and ½ hour to paint, and the Economy
model requires 2 hours to assemble and 1 hour to paint. The maximum number of
assembly hours available is 24 per day and the maximum number of painting hours
available is 8 per day. If the profit on the Deluxe model is $15 per unit and the profit on
the Economy model is $12 per unit, how many units of each model will maximize profit?
Let x = number of Deluxe models
y = number of Economy models
a. List the constraints
b. Determine the objective function. __________________
c. Graph the set of constraints. Place number of Deluxe models on the horizontal axis
and number of Economy models on the vertical axis.
d. Find the vertices of the feasible region.
Vertices Profit
e. How many Deluxe models and how many Economy models should the Wellbuilt
Company produce to maximize their profit?
Deluxe____________
Economy__________
f. What is the maximum profit? __________________
Using your definitions:
3x+2y ≤ 24 (assembly restraint) and
.5x+y ≤ 8 (painting restraint), alos
x ≥ 0 and y ≥ 0
Profit = 15x+12y
Solving the two corresponding equations above gave me (4,6)
so the vertices of your feasible region would be (0,0) (don't make any at all, silly)
(0,8) and (16,0)
"Sliding" the profit equation P=15x+12y, which has a slope of -5/4 away from the origin, while maintaining the slope would reach the point (4,6) of your region.
Can you tie all this together, and reach the correct answers?
Ok I understand the constratints, Can you please explain me how you got feasible region (0,0), (0,8) and (16,0).
And also slope -5/4.
Thanks for your help
hyuihi
a. The constraints for this problem are:
1) Assembly hours constraint: 3x + 2y ≤ 24
2) Painting hours constraint: 0.5x + y ≤ 8
3) Non-negativity constraint: x ≥ 0, y ≥ 0 (since we can't produce negative quantities)
b. The objective function is to maximize the profit, given by:
Profit = 15x + 12y
c. To graph the set of constraints, we can use a coordinate system with the number of Deluxe models (x) on the horizontal axis and the number of Economy models (y) on the vertical axis.
d. To find the vertices of the feasible region, we need to find the points of intersection of the lines given by the constraint equations:
3x + 2y = 24
0.5x + y = 8
Solving these equations will give us the vertices:
3x + 2y = 24 => y = (24 - 3x)/2
0.5x + y = 8 => y = 8 - 0.5x
Substituting the second equation into the first equation:
(24 - 3x)/2 = 8 - 0.5x
Simplifying the equation:
24 - 3x = 16 - x
2x = 8
x = 4
Substituting the value of x into one of the original equations:
0.5(4) + y = 8
2 + y = 8
y = 6
So the feasible region has vertices at (0,0), (4,6), and (8,0).
e. To find the number of Deluxe and Economy models that will maximize the profit, we evaluate the profit function at each of the vertices.
For (0,0):
Profit = 15(0) + 12(0) = 0
For (4,6):
Profit = 15(4) + 12(6) = 60 + 72 = 132
For (8,0):
Profit = 15(8) + 12(0) = 120
Therefore, the Wellbuilt Company should produce 4 Deluxe models and 6 Economy models to maximize their profit.
f. The maximum profit is $132.